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Properties

Label 2-31-31.30-c4-0-0
Degree $2$
Conductor $31$
Sign $-0.858 + 0.512i$
Analytic cond. $3.20446$
Root an. cond. $1.79010$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.08·2-s + 14.1i·3-s + 21.0·4-s + 3.89·5-s − 86.2i·6-s − 73.7·7-s − 30.8·8-s − 119.·9-s − 23.7·10-s − 69.3i·11-s + 298. i·12-s − 226. i·13-s + 449.·14-s + 55.1i·15-s − 149.·16-s + 439. i·17-s + ⋯
L(s)  = 1  − 1.52·2-s + 1.57i·3-s + 1.31·4-s + 0.155·5-s − 2.39i·6-s − 1.50·7-s − 0.482·8-s − 1.47·9-s − 0.237·10-s − 0.573i·11-s + 2.07i·12-s − 1.33i·13-s + 2.29·14-s + 0.245i·15-s − 0.582·16-s + 1.52i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 + 0.512i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $-0.858 + 0.512i$
Analytic conductor: \(3.20446\)
Root analytic conductor: \(1.79010\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 31,\ (\ :2),\ -0.858 + 0.512i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0496597 - 0.179989i\)
\(L(\frac12)\) \(\approx\) \(0.0496597 - 0.179989i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (825. - 492. i)T \)
good2 \( 1 + 6.08T + 16T^{2} \)
3 \( 1 - 14.1iT - 81T^{2} \)
5 \( 1 - 3.89T + 625T^{2} \)
7 \( 1 + 73.7T + 2.40e3T^{2} \)
11 \( 1 + 69.3iT - 1.46e4T^{2} \)
13 \( 1 + 226. iT - 2.85e4T^{2} \)
17 \( 1 - 439. iT - 8.35e4T^{2} \)
19 \( 1 + 115.T + 1.30e5T^{2} \)
23 \( 1 - 660. iT - 2.79e5T^{2} \)
29 \( 1 + 926. iT - 7.07e5T^{2} \)
37 \( 1 + 182. iT - 1.87e6T^{2} \)
41 \( 1 + 403.T + 2.82e6T^{2} \)
43 \( 1 - 3.46e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.88e3T + 4.87e6T^{2} \)
53 \( 1 - 924. iT - 7.89e6T^{2} \)
59 \( 1 - 2.45e3T + 1.21e7T^{2} \)
61 \( 1 + 515. iT - 1.38e7T^{2} \)
67 \( 1 - 2.07e3T + 2.01e7T^{2} \)
71 \( 1 - 1.98e3T + 2.54e7T^{2} \)
73 \( 1 + 5.41e3iT - 2.83e7T^{2} \)
79 \( 1 - 2.13e3iT - 3.89e7T^{2} \)
83 \( 1 - 415. iT - 4.74e7T^{2} \)
89 \( 1 - 9.13e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.11e4T + 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.78312611158152046444506407188, −15.94951341963879454839046136143, −15.20666454280855008727831096824, −13.12745286539427179825487727351, −11.01307413900698156679748294232, −10.06337265352104080236726196099, −9.499815017102116496345714201556, −8.171942843645534518849041452009, −5.98174526186838056241819185684, −3.49222981792196877283407031164, 0.21215932214832769026313400548, 2.13163248999879707494059160318, 6.65004453774282341337713341399, 7.19277346823000727926796057301, 8.849065345337045325523068318594, 9.869459134982427088407287433176, 11.64940867794923879616382814725, 12.81579958970484569288121462137, 13.94185484712389351248164873663, 16.10184865184493736557774758393

Graph of the $Z$-function along the critical line